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The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary-domain integral equation system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of a weak solutions for the nonlinear fractional elliptic systems with Dirichlet boundary conditions in three cases. We use the Leray–Schauder degree and some sufficient conditions for the solvability of a resonance and non‐resonance systems with respect to the spectrum of the fractional Laplacian. [ABSTRACT FROM AUTHOR]

A ℂ‐linear problem is considered for the Moisil‐Theodorescu system written in a complex form. The Fredholm theorem for this problem is proved, and the index formula obtained. [ABSTRACT FROM AUTHOR]

We study global gradient estimates for the weak solution to elliptic systems with measurable coefficients from composite materials in a nonsmooth bounded domain. The principal coefficients are assumed to be merely measurable in one variable and have small bounded mean oscillation (BMO) seminorms in the other variables on each subdomain whose boundary satisfies the so‐called δ‐Reifenberg flat condition. Under these assumptions and based on our new geometric observation for disjoint Reifenberg domains in a previous study, we obtain global W1,p estimates. [ABSTRACT FROM AUTHOR]

We prove optimal lower bounds for the growth of the potential energy over balls of minimizers to the vectorial Allen--Cahn energy in two spatial dimensions, as the radius tends to infinity. [ABSTRACT FROM AUTHOR]

We give some integrability conditions for the coefficients of a sequence of elliptic systems with varying coefficients in order to obtain the stability for homogenization. In the case of equations, it is well known that equi-integrability and bound in L1 are enough for this purpose; however, this is based on the maximum principle, and then, it does not work for systems. Here, we use an extension of the Murat-Tartar div-curl lemma because of M. Briane, J. Casado-Díaz, and F. Murat in order to obtain the stability by homogenization for systems uniformly elliptic, with bounded coefficients in [ABSTRACT FROM AUTHOR]

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n-th order nonlinear differential equations, equations with a generalized ϕ-Laplacian, and nonlinear partial differential systems. [ABSTRACT FROM AUTHOR]

This work deals with singular perturbation problems depending on small positive parameter ϵ. The limit problem as ϵ → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less-known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two-dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual-primal FETI methods as well as related inexact dual-primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI-DP approach and the behavior of FETI-DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual-primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems. [ABSTRACT FROM AUTHOR]

We show H1loc-regularity of the Cauchy stress tensor and H1loc-regularity of the infinitesimal strain tensor and the plastic strain tensor in infinitesimal Cosserat plasticity with monotone flow rule. We use energy estimates for difference quotients. [ABSTRACT FROM AUTHOR]

One successful approach in the design of solution methods for saddle-point problems requires the efficient solution of the associated Schur complement problem. In the case of problems arising from partial differential equations the factorization of the symbol of the operator can often suggest useful approximations for this problem. In this work we examine examples of preconditioners for regular elliptic systems of partial differential equations based on the Schur complement of the symbol of the operator and highlight the possibilities and some of the difficulties one may encounter with this approach. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For a generalized Stokes problem it is shown that weak solvability is equivalent to ellipticity of the system. In the case of ellipticity, the standard mixed finite element method converges if a Babuska-Brezzi condition for the pressure-form holds. This result is also true if the pressure operator is not the Lagrangean multiplier of the constraint. The results are applied to a non-isothermal gas flow in metalorganic chemical vapor deposition (MOCVD) reactors. Some numerical computations using Uzawa's method are given. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]